Exploration of New Classes of Bi-univalent Functions Defined by the Subordination Principle Using \(q\) -Gegenbauer Polynomials

In this study, we introduce a new class of bi-univalent functions constructed using \(q\) -Gegenbauer polynomials. We analyze and characterize this newly defined class of functions, and derive estimates for the Taylor-Maclaurin coefficients \(|a_{2}|\) and \(|a_{3}|\) . Additionally, we investigate the formulation of functional problems specific to functions within this subclass, namely \(\left| a_{3}-\sigma a_{2}^{2}\right| \) (commonly known as the Fekete-Szegö problem). By systematically exploring parameter specialization, we discover a range of novel outcomes that shed light on various aspects of our main results, contributing to a broader understanding of the mathematical landscape under investigation.